I’ve been away from campus visiting family during the first part of the summer, in one of those rare confluences of events where research, classes, and teaching all find themselves on temporary hiatus. While it’s pretty rare, I’m glad to take it. It is giving me Mathematica withdrawals though, especially with being able to easily make nice graphical plots and do some of the number-crunching I sometimes need for the more technical posts around here. Fortunately this is the last Sunday I’ll have that problem, as I’ll be back in Texas soon (and all those responsibilities will come roaring back). I’m looking forward to it.

For the moment I’d like to take this particular Sunday to kick off a Sunday Function project I’ve been thinking about for a while. Regular readers might recall the Greatest Physicists series here, and I’d like to do something similar with functions. Now obviously “Greatest Functions” is pretty much impossible. How could you even decide what counts as “one” function – are the exponential function and the trig functions different, or are they really different aspects of the same thing in complex analysis? What about polynomials and roots? I don’t think a ranking of importance going to work.

But there’s nothing stopping us from making a sort of Wonders of the World list for functions, unranked in any sense of “importance” but instead as a guide for tourists of mathematics to admire. Call it “10 Coolest Functions”. So just like the Greatest Physicists, it’s time to take nominations. Here’s the guidelines:

1. If it’s already been covered in Sunday Function, it’s still perfectly fine. I could probably do dozens of posts on (say) the sine function alone.

2. Anything an undergraduate in any major might encounter is fine. Try to avoid very obscure and/or technically difficult functions.

3. Relevance in mathematical physics is not required, but preferred.

4. Sets of functions are fine, in contexts like Bessel functions or Legendre polynomials.

5. Pick your favorite 10 (or less), and rank them if you’d like.

Remember that this is a loose and fun exercise; a listing of favorites and subjective coolness rather than by “importance”. You have lots of latitude, so if for instance you want to pick exponential/logarithm as one of your favorites that’s fine.

Well it’s a difficult task. But I know you’re up to it!